Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with a basis and structure constants are natural numbers. Similar to what one has for product of irreps. There are many analogies between conjugacy classes and irreps in particular see this article.
Tanaka-Krein duality states that group can be reconstructed from the tensor category of its representations which is semisimple for finite groups, and hence carries the same information as ring + basis of irreps.
Question: Can one reconstruct a group having (ring + basis) made of conjugacy classes ?
If not - what partial information (e.g. character table) one can get ?
Question: Is there any relation between this ring and ring of irreps of the same group ? or may be some other group ?
(Remark. For abelian group they are isomorphic.)
Question: Are there any further analogies between ring of irreps and conjugacy classes except mentioned in the paper cited above ?